A Subclass of Janowski Starlike Functions Involving Mathieu-Type Series

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived.

Modular iterated integrals associated with cusp forms

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension of higher-order modular forms which, in contrast to the standard higher-order forms, applies to general Fuchsian groups of the first kind and, as such, is of independent interest.

Fourier extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We establish weighted $$L^p$$ L p -Fourier extension estimates for $$O(N-k) \times O(k)$$ O ( N - k ) × O ( k ) -invariant functions defined on the unit sphere $${\mathbb {S}}^{N-1}$$ S N - 1 , allowing for exponents p below the Stein–Tomas critical exponent $$\frac{2(N+1)}{N-1}$$ 2 ( N + 1 ) N - 1 . Moreover, in the more general setting of an arbitrary closed subgroup $$G \subset O(N)$$ G ⊂ O ( N ) and G-invariant functions, we study the implications of weighted Fourier extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for G-invariant solutions to the nonlinear Helmholtz equation $$\begin{aligned} -\Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}({\mathbb {R}}^{N}), \end{aligned}$$ - Δ u - u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p ( R N ) , where Q is a nonnegative bounded and G-invariant weight function.

Резонансный отклик масштабно-инвариантных функций случайного процесса с турбулентным спектром

Scale-invariant random processes with large fluctuations are modeled by a system of two stochastic nonlinear differential equations describing interacting phase transitions. It is shown that under the action of white noise, a critical state arises, characterized by a turbulent spectrum and a scale-invariant distribution function. The critical state corresponds to the maximum entropy, which indicates the stability of the process. An external harmonic action on a random process with a turbulent spectrum gives rise to a resonant response of scale-invariant functions.

Of lions and Yakshis

Vladimir Propp’s theory Morphology of the Folktale identifies 31 invariant functions, subfunctions, and seven classes of folktale characters to describe the narrative structure of the Russian magic tale. Since it was first published in 1928, Propp’s approach has been used on various folktales of different cultural backgrounds. ProppOntology models Propp’s theory by describing narrative functions using a combination of a function class hierarchy and characteristic relationships between the Dramatis Personae for each function. A special focus lies on the restrictions Propp defined regarding which Dramatis Personae fulfill a certain function. This paper investigates how an ontology can assist traditional Humanities research in examining how well Propp’s theory fits for folktales outside of the Russian–European folktale culture. For this purpose, a lightweight query system has been implemented. To determine how well both the annotation schema and the query system works, twenty African tales and fifteen tales from the Kerala region in India were annotated. The system is evaluated by examining two case studies regarding the representation of characters and the use of Proppian functions in African and Indian tales. The findings are in line with traditional analogous Humanities research. This project shows how carefully modelled ontologies can be utilized as a knowledge base for comparative folklore research.

On equicontinuous factors of flows on locally path-connected compact spaces

We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

Divide and Invest: Bargaining in a Dynamic Framework

Many negotiations (for instance, among political parties or partners in a business) are characterized by dynamic bargaining: current agreements affect future bargaining possibilities. We study such situations using bargaining games á la Rubinstein (Econometrica 50:97–109, 1982), with the novelty that players can decide how much to invest, as well as how to share the residual surplus for their own consumption. Their investment decisions affect the size of the next surplus. In line with the existing literature, we focus on Markov Perfect Equilibria, where consumption and investment are linear time-invariant functions of capital and show that standard results in bargaining theory can be overturned. For instance, a more patient proposer may consume less than his opponent. The intuition is that when capital is productive, both parties have incentives to invest, however, the most patient party wishes to invest significantly more than his opponent. Then, to prioritize investment—which affects future bargaining possibilities—the former must make larger concessions and let the latter consume more. Another interesting result is that if a player becomes more patient, both parties may reduce their investment. The key underlying driver of this result is that when counteroffers become cheaper for an impatient party, he is able to reduce his investment and consume more. This forces his opponent to make larger concessions (and reduce his investment plan). Moreover, extreme demands (where a player consumes all the residual surplus) are possible in equilibrium, under fairly modest assumptions. Finally, only when bargaining is frictionless, is the equilibrium efficient.